A quantum transition can be seen as a result of interference between various pathways (e.g., Feynman paths), which can be labeled by a variable
f. An attempt to determine the value of
f without destroying the coherence between the pathways produces a weak value of
f. We show
f to be an average obtained with an amplitude distribution which can, in general, take negative values, which, in accordance with the uncertainty principle, need not contain information about the actual range of
f which contributes to the transition. It is also demonstrated that the moments of such alternating distributions have a number of unusual properties which may lead to a misinterpretation of the weak-measurement results. We provide a detailed analysis of weak measurements with and without post-selection. Examples include the double-slit diffraction experiment, weak von Neumann and von Neumann--like measurements, traversal time for an elastic collision, phase time, and local angular momentum.